The nature of opinion formation dynamics in complex networks is investigated using eigenmode analysis. Opinion formation dynamics is modeled by a decision-by-majority process of spin-like variables located at vertices of complex networks. Hamiltonian of the system is defined, and estimated by the eigenvalue and eigenvector of the adjacency matrix constructed from several network models. Then, the eigenmodes of initial and final state of the dynamics are analyzed by numerical studies. It is shown that the magnitude of the largest eigenvector at the initial states are key determinant for the resulting dynamics. It is proved that the final state of the dynamics can be estimated by the eigenmodes of the initial state.